“Not-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math>”, representation symmetry-protected topological, and Potts phases in an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>S</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math>-invariant chain
Edward E. O’Brien, Éric Vernier, Paul Fendley
Abstract
We analyze in depth an ${S}_{3}$-invariant nearest-neighbor quantum chain in the region of a $U(1)$-invariant self-dual multicritical point. We find four distinct proximate gapped phases. One has three-state Potts order, corresponding to topological order in a parafermionic formulation. Another has ``representation'' symmetry-protected topological (RSPT) order, while its dual exhibits an unusual ``not-$A$'' order, where the spins prefer to align in two of the three directions. Within each of the four phases, we find a frustration-free point with exact ground state(s). The exact ground states in the not-$A$ phase are product states, each an equal-amplitude sum over all states where one of the three spin states on each site is absent. Their dual, the RSPT ground state, is a matrix product state similar to that of Affleck-Kennedy-Lieb-Tasaki. A field-theory analysis shows that all transition lines are in the universality class of the critical three-state Potts model. They provide a lattice realization of a flow from a free-boson field theory to the Potts conformal field theory.